Proof by mathematical induction: Professional practice for secondary teachers

Mathematical induction is a proof technique that can be applied to establish the veracity of mathematical statements. This professional practice paper offers insight into mathematical induction as it pertains to the Australian Curriculum: Mathematics (ACMSM065, ACMSM066) and implications for how secondary teachers might approach this technique with students. In particular, literature on proof—and specifically, mathematical induction—will be presented, and several worked examples will outline the key steps involved in solving problems. After various teaching and learning caveats have been explored, the paper will conclude with some mathematical induction example problems that can be used in the secondary classroom

From Euclidean Geometry to Knots and Nets

This document is the Accepted Manuscript of an article accepted for publication in Synthese. Under embargo until 19 September 2018. The final publication is available at Springer via https://doi.org/10.1007/s11229-017-1558-x.This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if (a) it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, (b) the information thus displayed is not metrical and (c) it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions.Peer reviewe

Analysing Student Work Involving Geometric Concepts

Hyunyi Jung reflects on why students struggle to understand trigonometry

Holes in the Archive - to fill or to leave, that is the question...

The essay features in an issue of Bright Light dedicated to the idea of the substrate, and began as a paper in one of three symposia exploring the various interpretative investigations of the substrate, and how it impacts on, and informs different practices. Abstract: Holes in the archive - to fill or to leave, that is the question facing the researcher. The archive contains holes caused by lack of information. The dilemma of what to do when there is a missing link or a hole in evidence, generates new lines of thinking for knowledge production. Lacking hard facts leads to different research tactics, following hunches, reflexive and critical stock taking, asking questions and undertaking interviews. In conducting this process anecdotes surface. These accounts are often perceived to be trivial and of marginal research interest. In contradistinction, I assert the reverse, these hand-holds conjure the possibility of new animations. Penetrating the substrate or as I prefer 'substrata', is like plummeting the archive. This paper presents case studies from Peter Townsend's editorial archive of Studio International magazine where I have employed tactical and dynamic investigative processes

Bayesian chronological modeling of SunWatch, a fort ancient village in Dayton, Ohio

Radiocarbon results from houses, pits, and burials at the SunWatch site, Dayton, Ohio, are presented within an interpretative Bayesian statistical framework. The primary model incorporates dates from archaeological features in an unordered phase and uses charcoal outlier modeling (Bronk Ramsey 2009b) to account for issues of wood charcoal 14C dates predating their context. The results of the primary model estimate occupation lasted for 1–245 yr (95% probability), starting in cal AD 1175–1385 (95% probability) and ending in cal AD 1330–1470 (95% probability). An alternative model was created by placing the 14C dates into two unordered phases corresponding with horizontal stratigraphic relationships or distinct groups of artifacts thought to be temporally diagnostic. The results of the alternative model further suggest that there is some temporal separation between Group 1 and Group 2, which seems more likely in the event of a multicomponent occupation. Overall, the modeling results provide chronology estimates for SunWatch that are more accurate and precise than that provided in earlier studies. While it is difficult to determine with certainty if SunWatch had a single-component or multicomponent occupation, it is clear that SunWatch’s occupation lasted until the second half of the AD 1300s

Why 'scaffolding' is the wrong metaphor : the cognitive usefulness of mathematical representations.

The metaphor of scaffolding has become current in discussions of the cognitive help we get from artefacts, environmental affordances and each other. Consideration of mathematical tools and representations indicates that in these cases at least (and plausibly for others), scaffolding is the wrong picture, because scaffolding in good order is immobile, temporary and crude. Mathematical representations can be manipulated, are not temporary structures to aid development, and are refined. Reflection on examples from elementary algebra indicates that Menary is on the right track with his ‘enculturation’ view of mathematical cognition. Moreover, these examples allow us to elaborate his remarks on the uniqueness of mathematical representations and their role in the emergence of new thoughts.Peer reviewe